126 Chapter 2. Interaction of Radiation with Matter
An important factor is the direction of emission of Cherenkov light. As the light is
emitted in the form of a cone, we can define an angle of emission as the direction
of the cone. This angle Θcof the Cherenkov for a particle moving in a medium of
refractive indexnis given by
Θc= arccos(
1
βn)
. (2.5.10)
This equation can be used to define the maximum angle Θmaxc that one should
expect to see in a medium. The maximum will occur whenβ= 1, that is, when the
particle’s velocity is approximately equal to the velocity of lightin vacuum. Hence
Θmaxc = arccos(
1
n)
. (2.5.11)
Example:
Compute the maximum angle of the Cherenkov cone one can observe in water
having refractive index of 1.4.
Solution:
We use equation 2.5.11 to compute the desired angle.Θmaxc = arccos(
1
n)
= arccos(
1
1. 4
)
=0.775 radians
=0. 775 × 180
π=44. 40
The number of Cherenkov photonsdNemitted by a particle having chargeze(e
is the unit electrical charge) per unit lengthdxof the particle flight in a medium
having refractive indexnis given by
dN
dx
=2παz^2∫
1
λ^2[
1 −
1
n^2 β^2]
dλ. (2.5.12)Hereαis the fine structure constant andλis the wavelength of light emitted. It is
interesting to note that the threshold of Cherenkov depends not only on the velocity
of the particle but also on the refractive index of light in the medium. Since the
refractive index is actually dependent on the wavelength, therefore by just looking at
the refractive index we can deduce whether the photons of a particular wavelength
can be emitted or not. Based on this reasoning it can be shown that most of
the Cherenkov radiation emitted in water actually lies in the visible region of the
spectrum. Therefore the above equation can be safely integrated in the visible region
of the spectrum to get a good approximation of the number of photons emitted per
unit path length. This yields
dN
dx= 490z^2 sin^2 Θ cm−^1 (2.5.13)